Show that if A^2 = 0 then I-A in invertible and (I-A) ^-1=I+A.

Question

Wilson Merritt

Mathematics

27 March, 15:53

Show that if A^2 = 0 then I-A in invertible and (I-A) ^-1=I+A.

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Tristen · Answer 1

Let A be an nxn matrix. Show that if A^ (2) = 0, then I-A is nonsingular and (I-A) ^ (-1) = I+A

Note that (I - A) (I + A)

= I (I + A) - A (I + A)

= (I - A) - (A + A^2)

= I - A^2

= I - 0, since A^2 = 0

= I.

Hence, I - A is non singular with inverse I + A (since the inverse is unique when it does exist)